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# Formulas de calculo diferencial

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### Formulas de calculo diferencial

1. 1. Fórmulas de Cálculo Diferencial e Integral (Página 1 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M. Fórmulas de Cálculo Diferencial e Integral VER.6.8 Jesús Rubí Miranda (jesusrubim@yahoo.com) http://www.geocities.com/calculusjrm/ VALOR ABSOLUTO 1 1 1 1 si 0 si 0 y 0 y 0 0 ó ó n n k k k k n n k k k k a a a a a a a a a a a a a a ab a b a a a b a b a a = = = = ≥⎧ = ⎨ − <⎩ = − ≤ − ≤ ≥ = ⇔ = = = + ≤ + ≤ ∏ ∏ ∑ ∑ EXPONENTES ( ) ( ) / p q p q p p q q qp pq p p p p p p qp q p a a a a a a a a a b a b a a b b a a + − ⋅ = = = ⋅ = ⋅ ⎛ ⎞ =⎜ ⎟ ⎝ ⎠ = LOGARITMOS 10 log log log log log log log log log log ln log log ln log log y log ln x a a a a a a a r a a b a b e N x a MN M N M M N N N r N N N N a a N N N N N = ⇒ = + = − = = = = = = ALGUNOS PRODUCTOS ad+( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 3 3 2 2 3 3 3 2 2 3 2 2 2 2 2 2 3 3 3 3 2 2 2 a c d ac a b a b a b a b a b a b a ab b a b a b a b a ab b x b x d x b d x bd ax b cx d acx ad bc x bd a b c d ac ad bc bd a b a a b ab b a b a a b ab b a b c a b c ab ac bc ⋅ + = + ⋅ − = − + ⋅ + = + = + + − ⋅ − = − = − + + ⋅ + = + + + + ⋅ + = + + + + ⋅ + = + + + + = + + + − = − + − + + = + + + + + 1 1 n n k k n n k a b a ab b a b a b a a b ab b a b a b a a b a b ab b a b a b a b a b n− − = − ⋅ + + = − − ⋅ + + + = − − ⋅ + + + + = − ⎛ ⎞ − ⋅ = − ∀ ∈⎜ ⎟ ⎝ ⎠ ∑ ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 3 3 3 2 2 3 4 4 4 3 2 2 3 4 5 5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 3 3 3 2 2 3 4 4 4 3 2 2 3 4 5 5 5 4 3 2 2 3 4 5 6 6 a b a ab b a b a b a a b ab b a b a b a a b a b ab b a b a b a a b a b a b ab b a b + ⋅ − + = + + ⋅ − + − = − + ⋅ − + − + = + + ⋅ − + − + − = − ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 impar 1 par n k n k k n n k n k n k k n n k a b a b a b n a b a b a b n + − − = + − − = ⎛ ⎞ + ⋅ − = + ∀ ∈⎜ ⎟ ⎝ ⎠ ⎛ ⎞ + ⋅ − = − ∀ ∈⎜ ⎟ ⎝ ⎠ ∑ ∑ SUMAS Y PRODUCTOS n ( ) ( ) 1 2 1 1 1 1 1 1 1 1 0 n k k n k n n k k k k n n n k k k k k k k n k k n k a a a a c nc ca c a a b a b a a a a = = = = = = = − = + + + = = = + = + − = − ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 1 2 1 2 3 2 1 3 4 3 2 1 4 5 4 3 1 2 1 1 2 1 2 = 2 1 1 1 1 2 1 2 3 6 1 2 4 1 6 15 10 30 1 3 5 2 1 ! n k nn k k n k n k n k n k n k n a k d a n d n a l r a rl ar a r r k n n k n n n k n n n k n n n n n n n k n n k = − = = = = = = + − = + −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ + − − = = − − = + = + + = + + = + + − + + + + − = = ⎛ ⎞ =⎜ ⎟ ⎝ ⎠ ∑ ∑ ∑ ∑ ∑ ∑ ∏ ( ) ( ) 0 ! , ! ! n n n k k k k n n k k n x y x y k − = ≤ − ⎛ ⎞ + = ⎜ ⎟ ⎝ ⎠ ∑ ( ) 1 2 1 2 1 2 1 2 ! ! ! ! k n nn n k k k n x x x x x x n n n + + + = ⋅∑ CONSTANTES 9…3.1415926535 2.71828182846e π = = … TRIGONOMETRÍA 1 sen csc sen 1 cos sec cos sen 1 tg ctg cos tg CO HIP CA HIP CO CA θ θ θ θ θ θ θ θ θ θ θ = = = = = = = radianes=180π CA CO HIP θ θ sin cos tg ctg sec csc 0 0 1 0 ∞ 1 ∞ 30 1 2 3 2 1 3 3 2 3 2 45 1 2 1 2 1 1 2 2 60 3 2 1 2 3 1 3 2 2 3 90 1 0 ∞ 0 ∞ 1 [ ] [ ] sin , 2 2 cos 0, tg , 2 2 1 ctg tg 0, 1 sec cos 0, 1 csc sen , 2 2 y x y y x y y x y y x y x y x y x y x y x π π π π π π π π π ⎡ ⎤ = ∠ ∈ −⎢ ⎥ ⎣ ⎦ = ∠ ∈ = ∠ ∈ − = ∠ = ∠ ∈ = ∠ = ∠ ∈ ⎡ ⎤ = ∠ = ∠ ∈ −⎢ ⎥ ⎣ ⎦ Gráfica 1. Las funciones trigonométricas: sin x , cos x , tg x : -8 -6 -4 -2 0 2 4 6 8 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 sen x cos x tg x Gráfica 2. Las funciones trigonométricas csc x , sec x , ctg x : -8 -6 -4 -2 0 2 4 6 8 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 csc x sec x ctg x Gráfica 3. Las funciones trigonométricas inversas arcsin x , arccos x , arctg x : -3 -2 -1 0 1 2 3 -2 -1 0 1 2 3 4 arc sen x arc cos x arc tg x Gráfica 4. Las funciones trigonométricas inversas arcctg x , arcsec x , arccsc x : -5 0 5 -2 -1 0 1 2 3 4 arc ctg x arc sec x arc csc x IDENTIDADES TRIGONOMÉTRICAS 2 2 2 2 2 2 sin cos 1 1 ctg csc tg 1 sec θ θ θ θ θ θ + = + = + = ( ) ( ) ( ) sin sin cos cos tg tg θ θ θ θ θ θ − = − − = − = − ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) sin 2 sin cos 2 cos tg 2 tg sin sin cos cos tg tg sin 1 sin cos 1 cos tg tg n n n n n θ π θ θ π θ θ π θ θ π θ θ π θ θ π θ θ π θ θ π θ θ π θ + = + = + = + = − + = − + = + = − + = − + = ( ) ( ) ( ) ( ) ( ) sin 0 cos 1 tg 0 2 1 sin 1 2 2 1 cos 0 2 2 1 tg 2 n n n n n n n n π π π π π π = = − = +⎛ ⎞ = −⎜ ⎟ ⎝ ⎠ +⎛ ⎞ =⎜ ⎟ ⎝ ⎠ +⎛ ⎞ = ∞⎜ ⎟ ⎝ ⎠ sin cos 2 cos sin 2 π θ θ π θ θ ⎛ ⎞ = −⎜ ⎟ ⎝ ⎠ ⎛ ⎞ = +⎜ ⎟ ⎝ ⎠ ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 sin sin cos cos sin cos cos cos sin sin tg tg tg 1 tg tg sin 2 2sin cos cos2 cos sin 2tg tg2 1 tg 1 sin 1 cos2 2 1 cos 1 cos2 2 1 cos2 tg 1 cos2 α β α β α β α β α β α β α β α β α β θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ ± = ± ± = ± ± = = = − = − = − = + − = + ∓ ∓ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 sin sin 2sin cos 2 2 1 1 sin sin 2sin cos 2 2 1 1 cos cos 2cos cos 2 2 1 1 cos cos 2sin sin 2 2 α β α β α β α β α β α β α β α β α β α β α β α β + = + ⋅ − − = − ⋅ + + = + ⋅ − − = − + ⋅ − ( )sin tg tg cos cos α β α β α β ± ± = ⋅ ( ) ( ) ( ) ( ) ( ) ( ) 1 sin cos sin sin 2 1 sin sin cos cos 2 1 cos cos cos cos 2 α β α β α β α β α β α β α β α β α β ⋅ = − + +⎡ ⎤⎣ ⎦ ⋅ = − − +⎡ ⎤⎣ ⎦ ⋅ = − + +⎡ ⎤⎣ ⎦ tg tg tg tg ctg ctg α β α β α β + ⋅ = + FUNCIONES HIPERBÓLICAS sinh 2 cosh 2 sinh tgh cosh 1 ctgh tgh 1 2 sech cosh 1 2 csch sinh x x x x x x x x x x x x x e e x x e e x x e e e e x x e e x x e e x x e e − − − − − − − = + = − = = + + = = − = = + = = − x x e e− − x x [ { } ] { } { } sinh : cosh : 1, tgh : 1,1 ctgh : 0 , 1 1, sech : 0,1 csch : 0 0 → → ∞ → − − → −∞ − ∪ ∞ → − → − Gráfica 5. Las funciones hiperbólicas sinh x , cosh x , tgh x : -5 0 5 -4 -3 -2 -1 0 1 2 3 4 5 senh x cosh x tgh x FUNCIONES HIPERBÓLICAS INV ( ) ( ) 1 2 1 2 1 1 2 1 2 1 sinh ln 1 , cosh ln 1 , 1 1 1 tgh ln , 1 2 1 1 1 ctgh ln , 1 2 1 1 1 sech ln , 0 1 1 1 csch ln , 0 x x x x x x x x x x x x x x x x x x x x x x x x x − − − − − − = + + ∀ ∈ = ± − ≥ +⎛ ⎞ = <⎜ ⎟ −⎝ ⎠ +⎛ ⎞ = >⎜ ⎟ −⎝ ⎠ ⎛ ⎞± − ⎜ ⎟= < ≤ ⎜ ⎟ ⎝ ⎠ ⎛ ⎞+ ⎜ ⎟= + ≠ ⎜ ⎟ ⎝ ⎠
2. 2. Fórmulas de Cálculo Diferencial e Integral (Página 2 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M. IDENTIDADES DE FUNCS HIP 2 2 sinh 1x x ( ) ( ) ( ) 2 2 2 2 cosh 1 tgh sech ctgh 1 csch sinh sinh cosh cosh tgh tgh x x x x x x x x x x − = − = − = − − = − = − − = ( ) ( ) ( ) 2 2 2 sinh sinh cosh cosh sinh cosh cosh cosh sinh sinh tgh tgh tgh 1 tgh tgh sinh 2 2sinh cosh cosh 2 cosh sinh 2tgh tgh 2 1 tgh x y x y x y x y x y x y x y x y x y x x x x x x x x x ± = ± ± = ± ± ± = ± = = + = + ( ) ( ) 2 2 2 1 sinh cosh 2 1 2 1 cosh cosh 2 1 2 cosh 2 1 tgh cosh 2 1 x x x x x x x = − = + − = + sinh 2 tgh cosh 2 1 x x x = + cosh sinh cosh sinh x x e x x e x x− = + = − OTRAS ( ) ( ) 2 2 2 0 4 2 4 discriminante exp cos sin si , ax bx c b b ac x a b ac i e iα α β β β α β + + = − ± − ⇒ = − = ± = ± ∈ LÍMITES ( ) 1 0 0 0 0 1 lim 1 2.71828... 1 lim 1 sen lim 1 1 cos lim 0 1 lim 1 1 lim 1 ln x x x x x x x x x x e e x x x x x e x x x → →∞ → → → → + = = ⎛ ⎞ + =⎜ ⎟ ⎝ ⎠ = − = − = − = DERIVADAS ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 1 lim lim 0 x x x n n f x x f xdf y D f x dx x x d c dx d cx c dx d cx ncx dx d du dv dw u v w dx dx dx dx d du cu c dx dx ∆ → ∆ → − + ∆ − ∆ = = = ∆ ∆ = = = ± ± ± = ± ± ± = ( ) ( ) ( ) ( ) ( ) 2 1n n d dv du uv u v dx dx dx d dw dv du uvw uv uw vw dx dx dx dx v du dx u dv dxd u dx v v d du u nu dx dx − = + = + + −⎛ ⎞ =⎜ ⎟ ⎝ ⎠ = ( ) ( ) ( ) ( ) 12 1 2 (Regla de la Cadena) 1 donde dF dF du dx du dx du dx dx du dF dudF dx dx du x f tf tdy dtdy dx dx dt f t y f t = ⋅ = = =⎧′ ⎪ = = ⎨ ′ =⎪⎩ DERIVADA DE FUNCS LOG & EXP ( ) ( ) ( ) ( ) ( ) ( ) 1 ln log log log log 0, 1 ln ln a a u u u u v v v u dx u u dx d e du u dx u dx ed du u a dx u dx d du e e dx dx d du a a a dx dx d du dv u vu u u dx dx dx − = = ⋅ = ⋅ = ⋅ > = ⋅ = ⋅ = + ⋅ ⋅ 1d du dx du a ≠ DERIVADA DE FUNCIONES TRIGO ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 sin cos cos sin tg sec ctg csc sec sec tg csc csc ctg vers sen u u dx dx d du u u dx dx d du u u dx dx d du u u dx dx d du u u u dx dx d du u u u dx dx d du u u dx dx = = − = = − = = − = d du DERIV DE FUNCS TRIGO INVER ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 sin 1 1 cos 1 1 tg 1 1 ctg 1 si 11 sec si 11 si 11 csc si 11 1 vers 2 u dx dxu d du u dx dxu d du u dx dxu d du u dx dxu ud du u udx dxu u ud du u udx dxu u d du u dx dxu u ∠ = ⋅ − ∠ = − ⋅ − ∠ = ⋅ + ∠ = − ⋅ + + >⎧ ∠ = ± ⋅ ⎨ 1d du − < −⎩− − >⎧ ∠ = ⋅ ⎨ + < −⎩− ∠ = ⋅ − ∓ DERIVADA DE FUNCS HIPERBÓLICAS 2 2 sinh cosh cosh sinh tgh sech ctgh csch sech sech tgh csch csch ctgh u u dx dx d du u u dx dx d du u u dx dx d du u u dx dx d du u u u dx dx d du u u u dx dx = = = = − = − = − d du DERIVADA DE FUNCS HIP INV 1 2 -1 1 -12 1 2 1 2 1 1 12 senh 1 si cosh 01 cosh , 1 si cosh 01 1 tgh , 1 1 1 ctgh , 1 1 si sech 0, 0,11 sech si sech 0, 0,11 u dx dxu ud du u u dx dx uu d du u u dx u dx d du u u dx u dx u ud du u dx dx u uu u − − − − − − − = ⋅ + ⎧+ >± ⎪ = ⋅ > ⎨ − <− ⎪⎩ = ⋅ < − = ⋅ > − ⎧− > ∈⎪ = ⋅ ⎨ + < ∈− ⎩ ∓ 1d du 1 2 1 csch , 0 1 d du u u dx dxu u − ⎪ = − ⋅ ≠ + INTEGRALES DEFINIDAS, PROPIEDADES Nota. Para todas las fórmulas de integración deberá agregarse una constante arbitraria c (constante de integración). ( ) ( ){ } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) [ ] ( ) ( ) 0 , , , , si b b b a a a b b a a b c b a a c b a a b a a b a b b a a b b a a f x g x dx f x dx g x dx cf x dx c f x dx c f x dx f x dx f x dx f x dx f x dx f x dx m b a f x dx M b a m f x M x a b m M f x dx g x dx f x g x x a b f x dx f x dx a b ± = ± = ⋅ ∈ = + = − = ⋅ − ≤ ≤ ⋅ − ⇔ ≤ ≤ ∀ ∈ ∈ ≤ ⇔ ≤ ∀ ∈ ≤ < ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ INTEGRALES ( ) ( ) ( ) ( ) 1 Integración por partes 1 1 ln n n adx ax af x dx a f x dx u v w dx udx vdx wdx udv uv vdu u u du n n du u u + = = ± ± ± = ± ± ± = − = ≠ − + = ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ INTEGRALES DE FUNCS LOG & EXP ( ) ( ) ( ) ( ) ( ) ( ) 2 2 0 1ln 1 ln ln 1 ln ln ln 1 1 log ln ln 1 ln ln log 2log 1 4 ln 2ln 1 4 u u u u u u u u a a a e du e aa a du aa a ua du u a a ue du e u udu u u u u u u udu u u u u a a u u udu u u u udu u = >⎧ = ⎨ ≠⎩ ⎛ ⎞ = ⋅ −⎜ ⎟ ⎝ ⎠ = − = − = − = − = − = ⋅ − = − ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ INTEGRALES DE FUNCS TRIGO 2 2 sin cos cos sin sec tg csc ctg sec tg sec csc ctg csc udu u udu u udu u udu u u udu u u udu u = − = = = − = = − ∫ ∫ ∫ ∫ ∫ ∫ tg ln cos ln sec ctg ln sin sec ln sec tg csc ln csc ctg udu u u udu u udu u u udu u u = − = = = + = − ∫ ∫ ∫ ∫ ( ) 2 2 2 2 1 sin sin 2 2 4 1 cos sin 2 2 4 tg tg ctg ctg u udu u u udu u udu u u udu u u = − = + = − = − + ∫ ∫ ∫ ∫ sin sin cos cos cos sin u udu u u u u udu u u u = − = + ∫ ∫ INTEGRALES DE FUNCS TRIGO INV ( ) ( ) 2 2 2 2 2 2 sin sin 1 cos cos 1 tg tg ln 1 ctg ctg ln 1 sec sec ln 1 sec cosh csc csc ln 1 csc cosh udu u u u udu u u u udu u u u udu u u u udu u u u u u u u udu u u u u u u u ∠ = ∠ + − ∠ = ∠ − − ∠ = ∠ − + ∠ = ∠ + + ∠ = ∠ − + = ∠ − ∠ ∠ = ∠ + + − = ∠ + ∠ ∫ ∫ ∫ ∫ ∫ ∫ − INTEGRALES DE FUNCS HIP 2 2 sinh cosh cosh sinh sech tgh csch ctgh sech tgh sech csch ctgh csch udu u udu u udu u udu u u udu u u udu u = = = = − = − = − ∫ ∫ ∫ ∫ ∫ ∫ ( ) ( )1 tgh lncosh ctgh ln sinh sech tg sinh csch ctgh cosh 1 ln tgh 2 udu u udu u udu u udu u u − = = = ∠ = − = ∫ ∫ ∫ ∫ INTEGRALES DE FRAC ( ) ( ) 2 2 2 2 2 2 2 2 2 2 tg 1 ctg 1 ln 2 1 ln 2 du u a a a u a a du u a u a u a a u a du a u u a a u a a u = ∠ + = − ∠ − = > − + + = < − − ∫ ∫ ∫ 1 u INTEGRALES CON ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 sin cos ln 1 ln 1 cos 1 sec sen 2 2 ln 2 2 du u aa u u a du u u a u a du u au a u a a u du a a uu u a u a a u a u a u du a u a u a u a du u a u u a = ∠ − = −∠ = + ± ± = ± + ± = ∠ − = ∠ − = − + ∠ ± = ± ± + ± ∫ ∫ ∫ ∫ ∫ ∫ MÁS INTEGRALES ( ) ( ) 2 2 2 2 3 sin cos sin cos sin cos 1 1 sec sec tg ln sec tg 2 2 au au au e a bu b bu e bu du a b e a bu b bu e bu du a b udu u u u u − = + + = + = + + ∫ ∫ ∫ au ALGUNAS SERIES ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 0 0 0 0 0 0 0 2 2 3 3 5 7 2 1 1 2 4 6 '' ' 2! : Taylor ! '' 0 0 ' 0 2! 0 : Maclaurin ! 1 2! 3! ! sin 1 3! 5! 7! 2 1 ! cos 1 2! 4! nn n n n x n n f x x x f x f x f x x x f x x x n f x f x f f x f x n x x x e x n x x x x x x n x x x x − − − = + − + − + + = + + + + = + + + + + + = − + − + + − − = − + − ( ) ( ) ( ) ( ) ( ) 2 2 1 2 3 4 1 3 5 7 2 1 1 1 6! 2 2 ! ln 1 1 2 3 4 tg 1 3 5 7 2 1 n n n n n n x n x x x x x x n x x x x x x n − − − − − + + − − + = − + − + + − ∠ = − + − + + − −
3. 3. Fórmulas de Cálculo Diferencial e Integral (Página 3 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M. ALFABETO GRIEGO ayúscula Minúscula NombreM Equivalente Romano 1 Α α Alfa A 2 Β β Beta B 3 Γ γ Gamma G 4 ∆ δ Delta D 5 Ε ε Epsilon E 6 Ζ ζ Zeta Z 7 Η η Eta H 8 Θ θ ϑ Teta Q 9 Ι ι Iota I 10 Κ κ Kappa K 11 Λ λ Lambda L 12 Μ µ Mu M 13 Ν ν Nu N 14 Ξ ξ Xi X 15 Ο ο Omicron O 16 Π π ϖ Pi P 17 Ρ ρ Rho R 18 Σ σ ς Sigma S 19 Τ τ Tau T 20 Υ υ Ipsilon U 21 Φ φ ϕ Phi F 22 Χ χ Ji C 23 Ψ ψ Psi Y 24 Ω ω Omega W NOTACIÓN Seno.sin cos Coseno. tg Tangente. sec Secante. csc Cosecante. ctg Cotangente. vers Verso seno. arcsin sinθ θ= Arco seno de un ángulo θ . ( )u f x= sinh Seno hiperbólico. cosh Coseno hiperbólico. tgh Tangente hiperbólica. ctgh Cotangente hiperbólica. sech Secante hiperbólica. csch Cosecante hiperbólica. , ,u v w Funciones de x , , .( )u u x= ( )v v x= Conjunto de los números reales. { } Conjunto de enteros., 2, 1,0,1,2,= − −… … Conjunto de números racionales. c Conjunto de números irracionales. { }1,2,3,= … Conjunto de números naturales. Conjunto de números complejos.